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\frac{d}{dt}[G (\rho_\Lambda + \rho_m)] + 3 G H (\rho_m + p_m) = 0   \end{equation}  where $H$ is the Hubble rate $H = \frac{\dot{a}}{a}$ , and this equation implies the local conservation of matter   \begin{equation} \label{matterconservation} \frac{d}{dt}\rho_m + 3 G H (\rho_m + p_m) = 0   \end{equation}  only if $G$ and $\rho_\Lambda$ are constants, as in the standard model, or are both functions of time and satisfy the constraint  \begin{equation} \label{bianchiconstraint} (\rho_m + \rho_\Lambda) \frac{dG}{dt} + G \frac{d\rho_\lambda}{dt} = 0  \end{equation}  where $\rho_m$ is given by \ref{matterconservation}.